[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-272870-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-272870-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","worst-case-to-average-case-reduction-for-np","Worst-Case to Average-Case Reduction for NP","","The paper establishes a worst-case to average-case reduction for NP problems. It proves that if SAT is worst-case hard in the sense that NP does not have an efficient algorithm, then for every probabilistic polynomial-time algorithm attempting to decide SAT, there exists a polynomially samplable distribution on inputs where the algorithm fails with high probability. The work also refines this by showing a fixed quasi-polynomially samplable distribution hard for all such algorithms unless NP is easy in the worst case. The reduction relies on a lemma that generates few increasing-length Boolean formulas where the algorithm errs.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/worst-case-to-average-case-reduction-for-np/272870/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/worst-case-to-average-case-reduction-for-np/272870.png","ImageObject",300,407,{"name":42,"@type":43},"Aldword","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-11","2026-09-15",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",20,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What main reduction does the paper prove for NP?","Question",{"text":63,"@type":64},"It proves a worst-case to average-case reduction: if SAT is worst-case hard, then any probabilistic polynomial-time algorithm deciding SAT must fail with high probability on inputs drawn from some polynomially samplable distribution.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What does it mean for a distribution to be samplable in the paper?",{"text":68,"@type":64},"A distribution is samplable if a probabilistic polynomial-time machine can generate an ensemble of distributions where, on input 1^n, the output distribution equals D_n.",{"name":70,"@type":61,"acceptedAnswer":71},"Does the paper claim a single fixed distribution is hard for all algorithms?",{"text":72,"@type":64},"It does not rely on one fixed polynomially samplable distribution for all algorithms. Instead, it shows the existence of a fixed distribution on NP-complete language instances that is samplable in quasi-polynomial time and remains hard for all probabilistic polynomial-time algorithms unless NP is easy in the worst case.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},272870,1789468206,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":112,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":113,"slug":114},8,30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":55,"slug":118},9,"Religion & Spirituality","religion-spirituality",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":55,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":97,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":112,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":80,"read_time":144,"preview_page_count":4},2336478940917,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","IF NP LANGUAGES ARE HARD ON THE WORST-CASE, THEN IT IS EASY TO FIND THEIR HARD INSTANCES  \nDan Gutfreund, Ronen Shaltiel, and Amnon Ta-Shma  \nAbstract. We prove that if NP µ6 BPP, i.e., if SAT is worst-case hard, then for every probabilistic polynomial-time algorithm trying to decide SAT, there exists some polynomially samplable distribution that is hard for it. That is, the algorithm often errs on inputs from this distribution.  \nThis is the ¯rst worst-case to average-case reduction for NP of any kind.  \nWe stress however, that this does not mean that there exists one ¯xed samplable distribution that is hard for all probabilistic polynomial-time algorithms, which is a pre-requisite assumption needed for one-way functions and cryptography (even if not a su±cient assumption) . Nevertheless, we do show that there is a ¯xed distribution on instances of NPcomplete languages, that is samplable in quasi-polynomial time and is hard for all probabilistic polynomial-time algorithms (unless NP is easy in the worst case) .  \nOur results are based on the following lemma that may be of independent interest: Given the description of an e±cient (probabilistic) algorithm that fails to solve SAT in the worst case, we can e±ciently generate at most three Boolean formulae (of increasing lengths) such that the algorithm errs on at least one of them.  \nKeywords. Average-case complexity, Worst-case to average-case reductions, Foundations of cryptography, Pseudo classes  \nSubject classi¯cation. 68Q10 (Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q17 Computational di±culty of problems (lower bounds, completeness, di±culty of approximation, etc.) 94A60 Cryptography  \n2 Gutfreund, Shaltiel & Ta-Shma  \n1 . Introduction  \nIt is traditional in computational complexity to measure worst-case complexities, and say that a problem is feasible if it can be solved in worst-case polynomial time (i.e., it is in P or BPP) . A general belief is that NP-complete languages do not have feasible algorithms that are correct on every input. Thus under a worst-case measure of complexity, these problems are hard. However, this does not mean that in practice NP-complete problems are hard. It is possible that for a given problem, its hard instances are \\rare\", and in fact it is solvable e±ciently on all instances that actually appear in practice.  \n1.1. Average-case complexity: the cryptographic approach. This point of view led Levin (1986) to develop the theory of average-case complexity of NP problems, trying to capture the notions of easiness and hardness on the average. We now give a brief overview of the main concepts in this theory.  \nLet L be a language that we want to decide, and let D be a distribution over the instances. Here (and throughout the paper) we think of D as an ensemble of distributions fDngn2N, where Dn is a distribution over f0 ; 1gn.1 We want to measure how hard it is to decide L with respect to the distribution D. Thus we look at the complexity of the distributional problem (L; D) .  \nTrying to capture the notion of \\real-life\" instances, we look at input distributions that can be e±ciently generated. We say that D is samplable if there exists some probabilistic polynomial-time machine that generates the distribution. Formally, we require that there is a probabilistic polynomial time algorithm such that the output distribution of this algorithm on input 1n , is the distribution Dn.2  \nThis leads to a de¯nition of the distributional analogue of the class NP.  \nDefinition 1.1 . The class DistNP contains all the distributional problems (L; D), where L 2 NP and D is a samplable distribution.  \nLevin (1986) and subsequent papers (e.g. Ben-David et al. (1990); Gurevich (1990, 1991); Venkatesan & Levin (1988)) de¯ned the notions of reductions  \n1We mention that Levin considered a single distribution over all inputs, but Impagliazzo (1995)","cbCaibd7GQM06Oll","https://ap.wps.com/l/cbCaibd7GQM06Oll","pdf",274858,31,"English","# Introduction\n## Average-case complexity: the cryptographic approach\n# Preliminaries and definitions\n## DistNP and samplable distributions\n## Average-case hardness (Avgp)","[{\"question\":\"What main reduction does the paper prove for NP?\",\"answer\":\"It proves a worst-case to average-case reduction: if SAT is worst-case hard, then any probabilistic polynomial-time algorithm deciding SAT must fail with high probability on inputs drawn from some polynomially samplable distribution.\"},{\"question\":\"What does it mean for a distribution to be samplable in the paper?\",\"answer\":\"A distribution is samplable if a probabilistic polynomial-time machine can generate an ensemble of distributions where, on input 1^n, the output distribution equals D_n.\"},{\"question\":\"Does the paper claim a single fixed distribution is hard for all algorithms?\",\"answer\":\"It does not rely on one fixed polynomially samplable distribution for all algorithms. Instead, it shows the existence of a fixed distribution on NP-complete language instances that is samplable in quasi-polynomial time and remains hard for all probabilistic polynomial-time algorithms unless NP is easy in the worst case.\"}]","Worst-Case to Average-Case Reduction for NP | PDF",78]